I actually know it doesn't since if we use the integral test, and let $\ln(n+1)=u$ and $\displaystyle du=\frac{1}{n+1}dx$, then we have $$\int\frac{du}{u}\;,$$ which gives us $\ln(u)=\ln(\ln(n+1))$, which diverges. But, is there another way to find this?

I apologize for the format, not knowing Latex, I used WolframAlpha before to try to get the syntax, but they seemed to have made that a Pro account only feature now. I will make an effort to learn it in the future.